Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x+9y &= 9 \\ 5x+6y &= 6\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $6y = -5x+6$ Divide both sides by $6$ to isolate $y$ $y = {-\dfrac{5}{6}x + 1}$ Substitute this expression for $y$ in the first equation. $-5x+9({-\dfrac{5}{6}x + 1}) = 9$ $-5x - \dfrac{15}{2}x + 9 = 9$ Simplify by combining terms, then solve for $x$ $-\dfrac{25}{2}x + 9 = 9$ $-\dfrac{25}{2}x = 0$ $x = 0$ Substitute $0$ for $x$ back into the top equation. $-5( 0)+9y = 9$ $9y = 9$ $9y = 9$ $y = 1$ The solution is $\enspace x = 0, \enspace y = 1$.